A method for solar forecasting includes receiving a plurality of solar energy data as a function of time of day at a first time, forecasting from the solar energy data a mode, where the mode is a sunny day, a cloudy day, or an overcast day, and the forecast predicts the mode for a next solar energy datum, receiving the next solar energy datum, updating a probability distribution function (pdf) of the next solar energy datum given the mode, updating a pdf of the mode for the next solar energy datum from the updated pdf of the new solar energy datum given the mode, forecasting a plurality of future unobserved solar energy data from the updated pdf of the mode, where the plurality of future unobserved solar energy data and the plurality of solar energy data have a Gaussian distribution for a given mode determined from training data.
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1. A method for solar forecasting for managing a power grid, comprising the steps of: receiving a time series of solar energy data values as a function of time of day at a first time t; forecasting from the solar energy data a mode of day represented by the solar energy data, wherein the mode of day is one of a sunny day, a cloudy day, or an overcast day, and the forecast predicts the mode of day for a next solar energy datum; receiving a new solar energy datum corresponding to the next solar energy datum, and updating a probability distribution function (pdf) of the new solar energy datum given the mode of day; updating a pdf of the mode of day for the next solar energy datum from the updated pdf of the new solar energy datum given the mode of day; forecasting a plurality of future unobserved solar energy data values from the updated pdf of the mode of day, wherein the plurality of future unobserved solar energy data values y t and the time series of solar energy data values x t have a Gaussian distribution for a given mode of day s t =k determined from a set of training data.
3. The method of claim 2 , wherein forecasting a plurality of future unobserved solar energy data values comprises: evaluating a conditional probability P(y t |x 1:t ) of the plurality of future unobserved solar energy data values given the time series of solar energy data values from Σ k=1 K P(s t =k|x 1:t )P(y t |x t , s t =k), wherein P(s t =k|x 1:t ) is a conditional probability for the mode of day s t given the time series of solar energy data values x 1:t , P(y t |x t , s t =k) is a conditional probability of the plurality of future unobserved solar energy data values given the time series of solar energy data values x 1:t and the mode of day s t , wherein P(y t |x t , s t =k) is a Gaussian distribution with a mean {tilde over (m)} yx(k,t) =C yx(k,t) C xx(k,t) −1 (x t −m x(k,t) )+m y(k,t) and covariance {tilde over (C)} yx(k,t) =C yy(k,t) −C yx(k,t) C xx(k,t) −1 C xy(k,t) , and the sum is over all modes; and calculating the plurality of future unobserved solar energy data values {tilde over (y)} t from Σ k=1 K P(s t =k|x 1:t ){tilde over (m)} yx(k,t) .
4. The method of claim 2 , wherein determining the Gaussian distribution for a given mode of day s t =k from a set of training data comprises: receiving a set of vectors z of solar energy time series for N training days, each with T time stamps; classifying each day into a mode of day; splitting the set of vectors into sub-vectors according to the mode of day, wherein there is one sub-vector for each mode of day; determining the mean m k,t for each time stamp t from an average over all training days of mode k at time t; for each data point z t,k at a time t in a sub vector for mode k, forming a training set from τ data values that precede z t,k and from τ data values that succeed z t,k ; training a probabilistic principal component analyzer (PPCA) for each data point from the training set for each data point using eigen decomposition to determine matrices U k,t and Λ k,t , and a residual eigenvalue σ k,t 2 ; and initializing a categorical probability for the mode of day and a transition probability for the mode of day to predetermined values.
5. The method of claim 4 , wherein classifying each day into a mode of day comprises: classifying a time series value at time t as noisy if an absolute gradient at time t is greater than a first predetermined threshold; classifying a day as sunny if a percentage of noisy time series values is less than a second predetermined threshold; calculating an upper envelop for all sunny day time series values; defining a template for a cloudy day and a template for an overcast day from the sunny day envelop; matching a non-sunny day time series against each template using an Euclidean distance; and determining whether a non-sunny day is cloudy or overcast based on a smaller corresponding distance.
6. The method of claim 5 , wherein the mode of day includes a general mode trained on all available data for data values that fall between two modes.
7. The method of claim 4 , further comprising relearning the mean m k,t , the matrices U k,t and Λ k,t , the residual eigenvalue σ k,t 2 , and the transition probability for the mode of day using an expectation-maximization algorithm.
8. The method of claim 1 , wherein determining a mode of day from the solar energy data comprises evaluating a first conditional probability P(s t |X 1:t−1 ) of the mode of day s t for the next solar energy datum given the time series of solar energy data values x 1:t−1 from a product P(s t−1 |x 1:t−1 )P(s t |s t−1 ), a second conditional probability P(s t |x) is a categorical distribution, P(s t−1 |x 1:t−1 ) is a third conditional probability for a previous solar energy datum at time t−1,and P(s t |s t−1 ) is a transition probability of a mode of day changing between time t−1 and time t.
9. The method of claim 1 , wherein the probability distribution function (pdf) P(x t |s t ) of the new solar energy datum given the mode of day is a multivariate Gaussian distribution, and updating the pdf of the mode of day for the next solar energy datum from the updated pdf of the new solar energy datum given the mode of day comprises evaluating P(s t |x 1:t ) from P(s t |x 1:t−1 )P(x t |s t ).
11. The method of claim 10 , wherein classifying each day into a mode of day comprises: classifying a time series value of the solar energy time series at time t as noisy if an absolute gradient at time t is greater than a first predetermined threshold; classifying a day as sunny if a percentage of noisy time series values is less than a second predetermined threshold; calculating an upper envelop for all sunny day time series values; defining a template for a cloudy day and a template for an overcast day from the sunny day envelop; matching a non-sunny day time series against each template using an Euclidean distance; and determining whether a non-sunny day is cloudy or overcast based on a smaller corresponding distance.
12. The method of claim 10 , wherein determining a mode of day from the solar energy data comprises evaluating a first conditional probability P(s t |x 1:t−1 ) of the mode of day s t for the next solar energy datum given the time series of solar energy data values x 1:t−1 from a product P(s t−1 |x 1:t−1 )P(s t |s t−1 ), a second conditional probability P(s t |x) is a categorical distribution, P(S t−1 |x 1:t−1 )is a third conditional probability for a previous solar energy datum at time t−1,and P(s t |s t−1 ) is a transition probability of a mode of day changing between time t−1 and time t.
13. The method of claim 10 , wherein the probability distribution function (pdf) P(x t |s t ) of the new solar energy datum given the mode of day is a multivariate Gaussian distribution, and updating the pdf of the mode of day for the next solar energy datum from the updated pdf of the new solar energy datum given the mode of day comprises evaluating P(s t |x 1:t ) from P(s t |x 1:t−1 )P(x t |s t ).
14. The method of claim 10 , wherein forecasting a plurality of future unobserved solar energy data values comprises: evaluating a conditional probability P(y t |x 1:t ) of the plurality of future unobserved solar energy data values given the time series of solar energy data values from Σ k=1 K P(s t =k|x 1:t )P(y t |x t , s t =k), wherein P(s t =k|x 1:t ) is a conditional probability for the mode of day s t given the time series of solar energy data values x 1:t , P(y t |x t , s t =k)is a conditional probability of the plurality of future unobserved solar energy data values given the time series of solar energy data values x 1:t and the mode of day s t , wherein P(y t |x t , s t =k) is a Gaussian distribution with a mean {tilde over (m)} yx(k,t) C yx(k,t) C xx(k,t) −1 (x t −m x(k,t) )+m y(k,t) and covariance {tilde over (C)} yx(k,t) =C yy(k,t) −C yx(k,t) C xx(k,t) −1 C xy(k,t) , and the sum is over all modes; and calculating the plurality of future unobserved solar energy data values {tilde over (y)} t from Σ k=1 K P(s t =k|x 1:t ){tilde over (m)} yx(k,t) .
15. A non-transitory program storage device readable by a computer, tangibly embodying a program of instructions executed by the computer to perform the method steps for solar forecasting in order to manage a power grid, the method comprising the steps of: receiving a time series of solar energy data values as a function of time of day at a first time t; forecasting from the solar energy data a mode of day represented by the solar energy data, wherein the mode of day is one of a sunny day, a cloudy day, or an overcast day, and the forecast predicts the mode of day for a next solar energy datum; receiving a new solar energy datum corresponding to the next solar energy datum, and updating a probability distribution function (pdf) of the new solar energy datum given the mode of day; updating a pdf of the mode of day for the next solar energy datum from the updated pdf of the new solar energy datum given the mode of day; forecasting a plurality of future unobserved solar energy data values from the updated pdf of the mode of day, wherein the plurality of future unobserved solar energy data values y t and the time series of solar energy data values x t have a Gaussian distribution for a given mode of day s t =k determined from a set of training data.
17. The computer readable program storage device of claim 16 , wherein forecasting a plurality of future unobserved solar energy data values comprises: evaluating a conditional probability P(y t |x 1:t ) of the plurality of future unobserved solar energy data values given the time series of solar energy data values from Σ k=1 K P(s t =k|x 1:t )P(y t |x t , s t =k) wherein P(s t =k|x 1:t ) is a conditional probability for the mode of day s t given the time series of solar energy data values x 1:t , P(y t |x t , s t =k) is a conditional probability of the plurality of future unobserved solar energy data values given the time series of solar energy data values x 1:t and the mode of day s t , wherein P(y t |x t , s t =k) is a Gaussian distribution with a mean {tilde over (m)} yx(k,t) =C yx(k,t) C xx(k,t) −1 (x t −m x(k,t) )+m y(k,t) and covariance {tilde over (C)} yx(k,t) =C yy(k,t) −C yx(k,t) C xx(k,t) −1 C xy(k,t) , and the sum is over all modes; and calculating the plurality of future unobserved solar energy data values {tilde over (y)} t from Σ k=1 K P(s t =k|x 1:t ){tilde over (m)} yx(k,t) .
18. The computer readable program storage device of claim 16 , wherein determining the Gaussian distribution for a given mode of day s t =k from a set of training data comprises: receiving a set of vectors z of solar energy time series for N training days, each with T time stamps; classifying each day into a mode of day; splitting the set of vectors into sub-vectors according to the mode of day, wherein there is one sub-vector for each mode of day; determining the mean m k,t for each time stamp t from an average over all training days of mode k at time t; for each data point z t,k at a time t in a sub vector for mode k, forming the training set from τ data values that precede z t,k and from τ data values that succeed z t,k ; training a probabilistic principal component analyzer (PPCA) for each data point from training set for each data point using eigen decomposition to determine matrices U k,t and Λ k,t , and the residual eigenvalue σ k,t 2 ; and initializing a categorical probability for the mode of day and a transition probability for the mode of day to predetermined values.
19. The computer readable program storage device of claim 18 , wherein the method further includes relearning the mean m k,t , the matrices U k,t and Λ k,t , the residual eigenvalue σ k,t 2 , and the transition probability for the mode of day using an expectation-maximization algorithm.
20. The computer readable program storage device of claim 18 , wherein classifying each day into a mode of day comprises: classifying a time series value at time t as noisy if an absolute gradient at time t is greater than a first predetermined threshold; classifying a day as sunny if a percentage of noisy time series values is less than a second predetermined threshold; calculating an upper envelop for all sunny day time series values; defining a template for a cloudy day and a template for an overcast day from the sunny day envelop; matching a non-sunny day time series against each template using an Euclidean distance; and determining whether a non-sunny day is cloudy or overcast based on a smaller corresponding distance.
21. The computer readable program storage device of claim 20 , wherein the mode of day includes a general mode trained on all available data for data values that fall between two modes.
22. The computer readable program storage device of claim 15 , wherein determining a mode of day from the solar energy data comprises evaluating a first conditional probability P(s t |x 1:t−1 ) of the mode of day s t for the next solar energy datum given the time series of solar energy data values x 1:t−1 from a product P(s t−1 |x 1:t−1 )P(s t |s t−1 ), a second conditional probability P(s t |x)) is a categorical distribution, P(s t−1 |x 1:t−1 ) is a third conditional probability for a previous solar energy datum at time t−1, and P(s t |s t−1 ) is a transition probability of a mode of day changing between time t−1 and time t.
23. The computer readable program storage device of claim 15 , wherein the probability distribution function (pdf) P(x t |s t ) of the new solar energy datum given the mode of day is a multivariate Gaussian distribution, and updating the pdf of the mode of day for the next solar energy datum from the updated pdf of the new solar energy datum given the mode of day comprises evaluating P(s t |x 1:t ) from P(s t |x 1:t−1 )P(x t |s t ).
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June 29, 2015
August 20, 2019
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